Straight lines practice · Mixed difficulty

The angle between the lines represented by the equation $\left(x^2+y^2\right) \sin \theta+2 x y=0$ is

The orthocentre of the triangle formed by the lines $x=2, y=3$ and $3 x+2 y=6$ is at the point

The distance between the two points $A$ and $A^{\prime}$ which lie on $y=2$ such that both the line segments $A B$ and $A^{\prime} B$ (where $B$ is the point $(2,3)$ ) subtend angle $\frac{\pi}{4}$ at the origin, is equal to

Two lines are drawn through $(3,4)$, each of which makes angle of $45^{\circ}$ with the line $x-y=2$, then area of the triangle formed by these lines is $N$. Find $2 N$.

If the straight lines $x+3 y=4,3 x+y=4$ and $x+y=0$ form a triangle, then the triangle is

Let $A(1,1), B(6,5), C(3 / 2,2)$ be the vertices of $\triangle A B C$. If a point $P$ is such that areas of $\triangle P A B, \triangle P A C$ and $\triangle P B C$ are equal and $Q\left(-\frac{7}{6},-\frac{1}{3}\right)$ is a point, then $P Q=$

If a point $P\left( {4,\;3} \right)$ is shifted by a distance$\sqrt 2$ unit parallel to the line $y = x$, then coordinates of p in new position are

The equation of the sides of a triangle are $x - 3y = 0$, $4x + 3y = 5$ and $3x + y = 0$. The line $3x - 4y = 0$ passes through

The straight line $x+2 y-9=0,3 x+5 y-5=0$ and $a x+b y-1=0$ are concurrent if the straight line $35 x-22 y+1=0$ passes through the point

If $O$ is the origin and $Q$ is a variable point on $x^2=4 y$, then the locus of the mid-point of $O Q$, is

If the point (a,a) falls between the lines $\left| {x + y} \right| = 4$, then

The line which is parallel to x-axis and crosses the curve $y = \sqrt x \;$ at an angle $45^\circ$, is

The circumcentre of the triangle with vertices (0, 30), (4, 0) and (30, 0) is

The equation of a straight line parallel to $2 x+3 y+11=0$ and which is such that the sum of its intercept on the axes is 15 , is

Consider the following statements about straight lines:

I. Slope of horizontal line is zero and slope of vertical line is undefined.

II. Two lines are parallel if and only if their slopes are equal.

III. Two lines are perpendicular if and only if product of their slope is $-1 .$

Which of the above statements are true?

Let the base of a triangle lie along the line x=a and be of length 2a. The area of this triangle is ${a^2}$ if the vertex lies on the lines

The locus of a point represented by $x = \frac{a}{2}\left( {\frac{{t + 1}}{t}} \right),y = \frac{a}{2}\left( {\frac{{t - 1}}{t}} \right)\;$ is

The equation of a straight line passing through (1,2) and having intercept of length 3 between the straight lines $3x + 4y = 24$ and $3x\;4y = 12$ is

The line $x + y = 4$ divides the line joining $\left( { - 1,1} \right)$ and (5,7) in the ratio $\lambda :1$, then the value of $\lambda$ is

A line having slope '1' is drawn from a point $A(-3,0)$ cuts a curve $y=x^2+x+1$ at $P \,\&\, Q$. Find $\ell({AP}) \ell({AQ})$.

The line $x-2 y=0$ will be a bisector of the angle between the lines represented by the equation $x^2-2 h x y-2 y^2=0$, if $h=$

If a variable line drawn through the intersection of the lines $\frac{x}{3}+\frac{y}{4}=1$ and $\frac{x}{4}+\frac{y}{3}=1$, meets the coordinates axes at $A$ and $B$, then the locus of the mid-point of $A B$, is

Points $(1,2)$ and $(-2,1)$ are

A line passing through the point $\mathrm{P}(1,2)$ meets the line $x+y=7$ at the distance of 3 units from $P$. Then the slope of this line satisfies the equation:

If the line $px - qy = r$ intersects the coordinate axes at (a,0) and (0,b), thyen the value of a+b is equal to

A line through the point $A(2,4)$ intersects the line $x+y=9$ at the point $P.$ The minimum distance of $A P,$ is

If the lines $x + 3y - 9 = 0,\;4x + by - 2 = 0$ and $2x - y - 4 = 0$ are concurrent, then b equals

The straight lines $ax + by = c,\;bx + cy = a$ and $cx + ay = b$ are concurrent, if

The area enclosed by the lines $|x|+|y|=2$ is

The slopes of the lines which make an angle ${45^ \circ }$ with the line $3x - y = - 5$ are